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Multiplicity of solutions for non-local elliptic equations driven by the fractional Laplacian

Series:

CDSNS Colloquium

Tuesday, January 7, 2014 - 15:05

1 hour (actually 50 minutes)

Location:

Skiles 005

Speaker:

Xifeng Su

,

Beijing Normal University

We consider the semi-linear elliptic PDE driven by the fractional Laplacian:
\begin{equation*}\left\{%\begin{array}{ll} (-\Delta)^s u=f(x,u) & \hbox{in $\Omega$,} \\ u=0 & \hbox{in $\mathbb{R}^n\backslash\Omega$.} \\\end{array}%
\right.\end{equation*}An $L^{\infty}$ regularity result is given, using De Giorgi-Stampacchia iteration method.By
the Mountain Pass Theorem and some other nonlinear analysis methods,
the existence and multiplicity of non-trivial solutions for the above
equation are established. The validity of the Palais-Smale condition
without Ambrosetti-Rabinowitz condition for non-local elliptic equations
is proved. Two non-trivial solutions are given under some weak
hypotheses. Non-local elliptic equations with concave-convex
nonlinearities are also studied, and existence of at least six solutions
are obtained.
Moreover, a global result of
Ambrosetti-Brezis-Cerami type is given, which shows that the effect of
the parameter $\lambda$ in the nonlinear term changes considerably the
nonexistence, existence and multiplicity of solutions.